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Several algorithms based on depth-first search compute strongly connected components in linear time.. Kosaraju's algorithm uses two passes of depth-first search. The first, in the original graph, is used to choose the order in which the outer loop of the second depth-first search tests vertices for having been visited already and recursively explores them if not.
In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same ...
In this example, there is a true dependence between the j iteration of statement S1 and the j+1th statement of S1. This can be represented as S1[i,j] →T S1[i,j+1] The iteration space traversal graph and the loop carried dependence graph is: Iteration Space Traversal Graph: Loop Carried Dependence Graph:
Thus, a dynamic call graph can be exact, but only describes one run of the program. A static call graph is a call graph intended to represent every possible run of the program. The exact static call graph is an undecidable problem, so static call graph algorithms are generally overapproximations. That is, every call relationship that occurs is ...
The outermost loop searches each node that has not yet been visited, ensuring that nodes which are not reachable from the first node are still eventually traversed. The function strongconnect performs a single depth-first search of the graph, finding all successors from the node v, and reporting all strongly connected components of that subgraph.
Loop through the edges of the graph, in ascending sorted order by their weight. For each edge: Test whether adding the edge to the current forest would create a cycle. If not, add the edge to the forest, combining two trees into a single tree. At the termination of the algorithm, the forest forms a minimum spanning forest of the graph.
The same concept can be extended to multigraphs and graphs with loops by storing the number of edges between each two vertices in the corresponding matrix element, and by allowing nonzero diagonal elements. Loops may be counted either once (as a single edge) or twice (as two vertex-edge incidences), as long as a consistent convention is followed.
The key insight in the algorithm is as follows. If there is a cycle, then, for any integers i ≥ μ and k ≥ 0, x i = x i + kλ, where λ is the length of the loop to be found, μ is the index of the first element of the cycle, and k is a whole integer representing the number of loops.